Question

Three straight lines $L1$, $L2$ and $L3$ are parallel and lie in the same plane. A total of $m$ points are taken on $L1$, $n$ points on. $L2$, $k$ points on $L3$. The maximum number of triangles formed with vertices at these points are.

• A. ${}^{m+n+k}C_3$
• B. ${}^{m+n+k}C_3-{}^{m}C_3-{}^{n}C_3$
• C. ${}^{m+n+k}C_3+{}^{m}C_3+{}^{n}C_3$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

Find the maximum number of triangles:

In the question, it is given that three straight lines $L1$, $L2$ and $L3$ are parallel and lie in the same plane. A total of m points are taken on $L1$, $n$ points on. $L2$, $k$ points on $L3$.

The total number of points are $m+n+k$.

We know that we have to select $3$ point to form a triangle.

Therefore, the total number of ways in which we can select $3$ point are ${}^{m+n+k}C_3$.

Since to form a triangle all the three points cannot be on the same line.

So, the number of ways in which all the three points lie on the same plane can be given by: ${}^{m}C_3+{}^{n}C_3+{}^{k}C_3$.

The maximum number of triangles can be given by: ${}^{m+n+k}C_3-{}^{m}C_3-{}^{n}C_3-{}^{k}C_3$.

Therefore, The maximum number of triangles are ${}^{m+n+k}C_3-{}^{m}C_3-{}^{n}C_3-{}^{k}C_3$.

Hence, option $D$ is the correct option.